Optimal Crop Planning for Small-Scale Farmers using Portfolio Theory Considering Income Stability

Present Address:

Andrés Mauricio Paredes-Rodríguez 1✉ Emailandres.paredes@correounivalle.edu.co

Juan Pablo Orejuela-Cabrera 1 Emailjuan.orejuela@correounivalle.edu.co

Juan Carlos Osorio-Gómez 1 Emailjuan.osorio@correounivalle.edu.co

1 School of Industrial Engineering Universidad del Valle Cali ORCID Colombia

Andrés Mauricio Paredes-Rodríguez, School of Industrial Engineering, Universidad del Valle, Cali-Colombia, ORCID 0000-0001-9196-9965, andres.paredes@correounivalle.edu.co. Corresponding author.

Juan Pablo Orejuela-Cabrera, School of Industrial Engineering, Universidad del Valle, Cali – Colombia, ORCID 0000-0003-2187-0630, juan.orejuela@correounivalle.edu.co

Juan Carlos Osorio-Gómez, School of Industrial Engineering, Universidad del Valle, Cali - Colombia, ORCID 0000-0001-5625-5609, juan.osorio@correounivalle.edu.co

Abstract

Small-scale farmers play an indispensable role in achieving food sovereignty in developing countries, as they supply a significant share of food along the agricultural value chain. One of the most critical decisions they face is crop planning. Because agricultural production follows seasonal cycles, farmers often experience periods with limited or no income, which can severely affect their financial stability. This research proposes a novel mathematical model to determine the optimal combination of crops for small-scale farmers using a portfolio theory approach. The model seeks to maximise income while ensuring income stability by explicitly accounting for risks and uncertainties related to price variability and production cycles. The methodology consists of three phases. First, agricultural products suitable for cultivation in rural areas of a Colombian city are characterised based on technical, economic, and contextual criteria. Second, the identified crops are prioritised using the TOPSIS multi-criteria decision-making method. Finally, a commitment-based mixed-integer linear programming model is developed using assumptions derived from the characterisation phase and monthly price scenarios. The results show that crop portfolios focused exclusively on risk minimisation led to highly conservative strategies with limited diversification, whereas income-maximising portfolios increase exposure to volatility. In contrast, the balanced scenario identifies diversified crop portfolios that stabilise monthly income while maintaining acceptable profitability. These findings demonstrate the practical value of the proposed model as a decision-support tool for small-scale farmers and agricultural advisors, enabling informed crop planning decisions that enhance income stability and reduce vulnerability to market fluctuations.

Keywords

Crop planning

small-scale farmers

agrifood supply chains

optimization

portfolio theory.

1. Introduction

Agri-food supply chains involve various actors from production to food delivery to the final customer, including farmers, processing plants, wholesalers and retailers [1]. In developing countries, the management of product, information, and financial flows in agrifood supply chains, particularly those involving fresh products, faces greater challenges than other types of networks. The short shelf life of these goods, combined with limited logistics infrastructure in rural areas [2] the high density of production and commercialisation nodes [3], and the physical distance between them, often hinders communication and coordination. These factors, together with the variability of supply and demand [4], increase the likelihood of food losses and, consequently, economic losses, thereby heightening the vulnerability of the entire supply chain and undermining both food availability and the income of the actors involved [5].

In Colombia, these challenges are further exacerbated by the fragmented structure of the production stage of fresh food supply chains, particularly fruits and vegetables. According to [6], the country has 2,042,033 Agricultural Production Units (UPAs), of which 1,924,257 belong to individuals, and nearly half (48.3%) cultivate less than three hectares. his production structure reflects the predominance of small-scale farmers, whose limited land size, dispersed geographical distribution, and low production volumes constrain their capacity to ensure a stable and continuous supply of fresh food. These structural constraints have direct implications for national food security. Evidence from the Food Insecurity Experience Scale (FIES), as implemented by DANE within Colombia’s National Quality of Life Survey, indicates that 25.5% of Colombian households experienced moderate or severe food insecurity in 2024 [7]. The vulnerability of small-scale agricultural producers therefore not only threatens their own livelihoods but also undermines the resilience and reliability of domestic food supply systems.

For many of these farmers, agricultural work represents not only their primary source of income but also the most accessible—and frequently the only—form of employment available to them, making their economic stability and survival directly dependent on this activity. In Colombia, this dependence is reflected in national employment figures: according to the National Administrative Department of Statistics [7], rural employment reached approximately 3.69 million people in October, of whom 42.3% were engaged in agriculture, livestock farming, hunting, forestry, and fishing. However, despite the sector’s central role in sustaining rural livelihoods, farmers continue to face persistent challenges in production, harvesting, and distribution processes. These challenges are further exacerbated by limited resources, restricted access to financing, and insufficient government support, which together hinder effective planning and decision-making and ultimately threaten their livelihoods [8].

Within this context of fragmented production systems, limited financial resources, and heightened food security risks, one of the most critical planning decisions that small-scale farmers must make to ensure their long-term survival is crop planning [9]–[12]. Crop planning is particularly critical because it involves selecting which crops to cultivate while accounting for harvest timing and seasonality—factors that directly influence market prices and revenue outcomes.

For small-scale producers, crop planning is especially consequential, as it determines not only the crops, they will harvest in the future but also the level and stability of income available throughout the planning horizon. This income must be sufficient to sustain agricultural operations and to meet household basic needs, thereby linking crop choice directly to income generation and, ultimately, to farmers’ livelihoods and survival [13]. Ensuring a stable flow of income across successive production cycles is particularly challenging given smallholders’ limited access to financial resources, which restricts their capacity to absorb shocks, manage risk, and plan effectively. Consequently, crop choice transcends a purely economic calculation and constitutes a fundamental survival strategy for small-scale farmers.

Building on this understanding of crop choice as a survival-oriented decision, crop planning has increasingly been examined through the lens of portfolio theory, which provides a rigorous framework for balancing expected returns and risk under uncertainty. The application of portfolio theory to agricultural decision-making has been widely explored in the literature, including approaches that rely on variance-based risk measure [14] and value-at-risk indicators [15]. However, despite this growing body of literature, existing studies tend to prioritise income maximisation or risk reduction in isolation, paying limited attention to the trade-off between income maximisation and income stability in contexts characterised by high vulnerability, limited financial buffers, and strong dependence on agricultural income—conditions that are particularly salient for small-scale farmers. In such contexts, income volatility itself constitutes a critical source of risk, as sharp fluctuations in earnings can directly compromise farmers’ ability to meet basic household needs and maintain production continuity, thereby threatening their long-term survival.

Motivated by this gap, the present study examines crop planning as a portfolio optimisation problem that explicitly accounts for both expected income and income volatility. The study addresses the following research questions: (RQ1) How can crop planning decisions be modelled to jointly maximise expected income and minimise the risk associated with income volatility for small-scale farmers? (RQ2) How does explicitly incorporating income volatility as a decision criterion affect optimal crop portfolio composition compared to traditional income-maximising approaches?

This research makes three main contributions. First, it extends the application of portfolio theory in agricultural economics by framing income volatility not merely as a statistical property but as a key determinant of smallholder survival. Second, it proposes a novel optimisation framework that simultaneously maximises expected income and minimises income volatility, capturing the inherent trade-offs faced by small-scale farmers under uncertainty. Third, the model provides decision-relevant insights with direct implications for risk management and agricultural policy, offering a tool to design crop portfolios that enhance income stability and, consequently, the resilience and survival of small-scale farming households.

The remainder of this paper is organised as follows. Section 2 provides a review of the literature on crop planning, with particular emphasis on the objectives considered and the main methodological approaches adopted in previous studies. Section 3 presents the methodology employed in this research, while Section 4 reports the main results. Section 5 discusses the principal contributions and limitations of the study. Section 6 outlines the theoretical and practical implications of the findings, and Section 7 concludes the paper by summarising the main conclusions and highlighting directions for future research.

2. Literature review

To identify studies addressing the crop planning problem, a systematic and structured literature search was conducted across major academic databases, including Scopus and Web of Science. The search strategy employed key terms such as “crop planning” and was restricted to peer-reviewed research articles published within the last twelve years, ensuring both relevance and methodological currency. Table 1 summarises the main findings derived from this review, highlighting the methodological approaches adopted in the literature and the objectives considered in each study.

Table 1

Summary of crop planning optimisation models (2014–2025): objectives, methodologies, and constraints
Authors	Methodology	Objectives								Constraints
Authors	Methodology	Net Profit	Variance of net profit (risk measure)	Conditional Value-at-Risk	Income's MAD (Risk measure)	Water use	Crop diversity	Production Total	Cost	Water supply	Food and nutrition security	Available land	Crop diversification	Production capacity	Demand	Crop rotation	Land Usage Expansion	Government subsidy	Budget	Environmental constraints
[14]	Multi objective programming	X	X									X		X					X	X
[16]	Integer Linear programming	X										X	X	X		X
[15]	Integer Linear programming	X		X								X		X
[17]	Integer Linear programming	X										X		X	X
[18]	Multi objective programming	X				X		X		X		X		X
[19]	Integer Linear programming	X							X			X					X	X
[20]	Integer Linear programming	X								X		X	X	X		X
[21]	Multi objective programming	X				X				X		X		X
[22]	Stochastic Programming	X								X	X	X	X
[23]	Multi objective programming	X					X					X	X	X		X
[24]	Integer Linear programming	X										X	X			X
Our research	Multi objective programming	X			X							X		X

Despite extensive research on crop planning optimisation, the literature between 2014 and 2025 (see Table 1) has largely focused on profit maximisation under technical, environmental, or policy constraints. However, explicit modelling of income risk—measured through variance, downside risk, or CVaR—remains largely unexplored. Furthermore, food and nutrition security considerations are rarely incorporated as explicit constraints, and the economic vulnerability of small-scale farmers facing income volatility has received limited attention. This reveals a clear research gap in developing crop planning models that jointly address profitability, income stability, and farmer survival under real-world constraints.

From the perspective of smallholders, the decision of which crops to cultivate extends well beyond the expected income per hectare. Farmers must also consider factors such as their experience in managing specific crops [25]. yield levels, harvest cycles, and, critically, the variability of output prices and production costs. These elements shape the risk profile of each crop and may force farmers to trade off higher expected returns against greater income volatility. In practice, this often implies choosing between crops with potentially higher but unstable returns and those offering more predictable, albeit lower, income streams. Consequently, crop planning emerges as an inherently multi-criteria decision problem in which profitability, risk exposure, and income stability interact simultaneously [26].

Due to its inherently multi-objective nature, the crop planning problem has been addressed through a wide range of methodological approaches. Deterministic and multi-objective optimisation models—typically formulated as linear or mixed-integer programming—have predominantly focused on maximising expected profit, yield, or land-use efficiency, while incorporating sustainability and policy objectives through technical or regulatory constraints [18], [23], [24], [27]. Although these models provide valuable insights into optimal resource allocation, they generally treat income as a deterministic or expected-value outcome, offering limited consideration of income variability or downside risk.

Fuzzy and multi-criteria decision-making approaches have extended this framework by explicitly recognising uncertainty and conflicting objectives, particularly in relation to water use efficiency, environmental impacts, and perceived fairness among stakeholders [21], [28], [29]. However, in most cases, uncertainty is handled through imprecise goals or membership functions rather than through explicit risk measures capturing income volatility, leaving the stability of farmers’ earnings largely unaddressed.

Heuristic and metaheuristic algorithms have been employed to tackle the computational complexity and non-linearity of large-scale and dynamic agricultural systems [30]–[32]. While these methods enhance solution efficiency and scalability, they typically optimise deterministic or scenario-based objective functions, without explicitly modelling income risk or its implications for farmers’ economic resilience.

More recently, data-driven and learning-based approaches integrating big data analytics, machine learning, and deep reinforcement learning have emerged as powerful tools for predictive and adaptive decision-making under uncertainty, particularly in yield forecasting and soil–water interactions [28], [33], [34]. Despite their ability to capture complex dynamics, these approaches remain largely focused on prediction accuracy or short-term optimisation, rather than on embedding income stability or risk mitigation as explicit decision criteria within crop planning models.

In contrast to these strands of the literature, this research proposes a commitment-based mixed-integer linear programming model that explicitly balances income maximisation with income stability by minimising the risk associated with price volatility. The model adopts the mean absolute deviation (MAD) of income as a risk measure [35], adapted to the Colombian smallholder context where stable cash flows are essential for meeting both production-related and household commitments. By explicitly levelling income across the planning horizon and accounting for period-to-period variability, the proposed approach addresses a critical gap in the literature: the integration of income risk management into crop planning decisions as a determinant of smallholder survival.

To operationalise this risk-aware approach, the proposed model incorporates monthly price scenarios over the planning horizon, capturing the effects of seasonality, supply–demand dynamics, and harvest cycles on market prices. Within this framework, mean absolute deviation (MAD) is adopted as the risk measure, as it enables a robust trade-off between income maximisation and income stability while remaining computationally tractable and data efficient. This characteristic is particularly important in the context of small-scale farmers in Colombia, where access to detailed historical data and advanced computational resources is often limited.

Beyond identifying an optimal crop mix, the model is designed as a decision-support tool to assist farmers in managing income uncertainty arising from price volatility and external shocks. By promoting crop portfolio diversification across the planning horizon, the model helps reduce exposure to severe income fluctuations caused by market instability, adverse weather conditions, or other unforeseen events. At the same time, its outputs can support local authorities and extension services in advising smallholders on land use and resource allocation strategies. In this broader perspective, the proposed approach contributes to enhancing the economic stability of farming activities, safeguarding rural employment, and strengthening local food security by supporting more resilient and sustainable crop planning decisions.

3. Methodology

The methodology employed in this research adopts a quantitative approach utilising a range of methods including surveys, mathematical modelling and other engineering techniques to design a crop portfolio that ensures income stability for small-scale farmers while minimising their risk exposure. Figure 1 illustrates the methodological framework employed in the study, which is based on the IDEF0 methodology [36]. The framework primarily comprises three phases.

Fig. 1

Methodological framework for the development of the study

In the first stage, a characterisation of the small-scale farmers under study is conducted through consultation with primary and secondary information sources, thereby defining the available crops for planting in the region and gathering relevant data about them, which will subsequently be utilised in the following stages of the study. In the second phase, crops are prioritised using a multi-criteria tool known as TOPSIS. Finally, in the third stage, a mixed integer linear programming model based on commitments is developed considering the crops selected in the second stage and considering the concepts of portfolio theory presented by [35]. Each of the methodological phases is outlined in greater detail below.

Phase 1. Characterization of the farmer under study

In this stage, the data required for the research is gathered through surveys conducted in the rural area under study. This data is supplemented with information from technical reports and reports and government agency publications, to define a crop portfolio suitable for planting in the region, including details on costs, prices and yields per hectare.

Phase 2. Prioritization of crops using a multicriteria tool

In this phase, the crops identified in the first stage serve as the alternatives to be evaluated and a set of criteria is established to prioritise them. These criteria consider attributes such as production, yield per hectare, price paid for the product, costs and the small farmer's preference for each crop, with the latter data being obtained through the surveys conducted in the initial stage of the study. Once the alternatives and criteria are defined, the TOPSIS methodology is applied to carry out the corresponding prioritisation.

TOPSIS is a well-established multi-criteria decision-making technique that has been widely applied in agrifood supply chain contexts, including supplier selection [37], risk assessment [38] and sustainability measurement [39]. Its continued use in recent studies reflects its ability to support decisions involving multiple and potentially conflicting criteria in complex agricultural systems.

In the specific context of crop planning, TOPSIS has been increasingly employed within hybrid decision-making frameworks. For example, [40] integrate TOPSIS, PROMETHEE, and the Borda rule to support crop selection based on criteria such as production costs, market prices, labour requirements, customer demand, and pest risks. Similarly, [41] combine fuzzy AHP with TOPSIS and PROMETHEE to rank alternative crops according to economic, technical, social, and environmental indicators, with the objective of identifying crops with higher profit potential.

Along the same lines, [42] apply TOPSIS in combination with Shannon and Condorcet indices to prioritise crops whose supply chains exhibit higher potential for circular economic development. In their framework, TOPSIS is not used as a methodological innovation but rather as an interpretable aggregation and ranking mechanism that translates multiple performance indicators into actionable insights for decision-makers. This application highlights the role of TOPSIS as a practical decision-support tool within hybrid frameworks, particularly when the objective is to facilitate operational decision-making.

Although TOPSIS is not a novel technique and more advanced MCDM methods have been proposed in recent years, its selection in this study is deliberate. Compared to more complex approaches, TOPSIS offers a high degree of transparency, low computational burden, and ease of interpretation—features that are especially relevant in rural contexts characterised by limited data availability and the involvement of non-technical stakeholders, such as small-scale farmers and local authorities. More sophisticated MCDM methods often require extensive preference elicitation, parameter tuning, or large datasets, which may limit their practical applicability in such settings.

In the proposed framework, TOPSIS complements the optimisation model by transforming its quantitative outputs—such as income levels, income stability, and resource use—into an intuitive ranking of crop portfolios. By evaluating the relative closeness of each alternative to an ideal solution, TOPSIS supports the comparison of trade-offs between profitability and income stability while maintaining interpretability and operational relevance. Consequently, its adoption prioritises practical applicability and policy relevance over methodological novelty, aligning with the decision-support objectives of this research.

The TOPSIS methodology according to [43] is presented in Appendix A.

Phase 3. Definition of crop portfolio through the mathematical model

Portfolio theory, as originally formulated by Markowitz (see [44]), has long been applied in contexts where decisions involve a trade-off between expected returns and the risks arising from uncertainty [45]. In this regard, optimisation provides a natural framework for addressing problems grounded in portfolio theory, as demonstrated in more recent contributions [46].

In Markowitz’s portfolio theory, uncertainty in returns and the associated risk are typically measured by the variance of the portfolio, a classical statistical measure of the dispersion of a random variable around its mean. Moreover, variance is expressed through a quadratic function. Although convex Quadratic Programming (QP) models are nowadays computationally tractable, Linear Programming (LP) formulations remain considerably more attractive from a computational standpoint, particularly when integer or binary decision variables are required in the model [15], [47]

In the literature, several alternatives to variance have been proposed for measuring the dispersion of a random variable. One such measure is the Mean Absolute Deviation (MAD), which, unlike variance, relies on absolute rather than squared deviations [48]. When returns are discretised MAD and other risk measures have the advantage of being computable through Linear Programming [47]. Building on this rationale, [35] were the first to formally promote MAD as an alternative risk measure within portfolio theory, advocating it as a modelling choice rather than a mere computational simplification. By replacing variance with MAD, the optimisation problem becomes linear, reducing computational costs while preserving the rationale of risk as dispersion and offering a practical counterpart to the classical Markowitz framework [46]

While the use of MAD as a risk measure does not constitute a theoretical novelty, its application to the agricultural context—particularly in the design of crop portfolios for smallholder farmers—provides a distinctive contribution. In this setting, producers face risks not only from market volatility but also from climatic variability, land-use constraints, and limited access to financial instruments for hedging. The translation of a financial portfolio framework into this domain therefore represents an applied innovation: it operationalises optimisation approach in a context where decision-making tools are scarce yet urgently needed to improve income stability and resilience. Thus, the value of this work lies in demonstrating how a well-established theoretical framework can be meaningfully adapted to address pressing problems in small-scale agriculture.

It should be noted that, although other approaches have been employed to incorporate risk into crop-planning decisions—such as the use of CVaR and stochastic optimisation to address different risk scenarios (see [15], [49])—the adoption of MAD, as proposed by Konno and Yamazaki, leads to a purely Linear Programming formulation. This is considerably more manageable from a computational perspective, which is essential in the context of small-scale farmers in rural Colombia, where data availability is limited, and practical applicability outweighs the benefits of modelling extreme tail risk. Consequently, MAD provides a pragmatic and robust framework that balances theoretical rigour with the need for operational feasibility.

The mean absolute deviation (MAD) defined in Konno and Yamasaki's portfolio theory is defined in Equations 10 and 11.

$\:MAD=E\left[\left|\sum\:_{j}{R}_{j}*{X}_{j}-E\left[\sum\:_{j}{R}_{j}*{X}_{j}\right]\right|\right]\:\:\:\:\:\:\:\:\:\left(10\right)$

$\:MAD=\frac{1}{T}\sum\:_{t}\left|\sum\:_{j}\left({r1}_{jt}-{r}_{j}\right)*{X}_{j}\right|\:\:\:\:\:\:\:\:\:\left(11\right)$

Where

$\:{R}_{j}$

represents the rate of return of stock j,

$\:{X}_{j}$

is the percentage of the portfolio allocated to stock j,

$\:{r1}_{jt}$

is the return of stock j in period t and

$\:{r}_{j}$

is the average of the returns of the stock j over the entire observation period. If

$\:{a}_{jt}\:$

is considered as the distance between the return of stock j in period t with respect to the average returns of that stock (

$\:{a}_{jt}=$

$\:{r1}_{jt}-{r}_{j}$

), Eq. 12 is obtained.

$\:MAD=\frac{1}{T}\sum\:_{t}\left|\sum\:_{j}{a}_{jt}*{X}_{j}\right|\:\:\:\:\:\:\:\:\:\left(12\right)$

Building on this foundation, this research develops a commitment-based Linear Programming model grounded in the portfolio theory of Konno and Yamazaki. The model is designed to support fruit and vegetable farmers in a rural Colombian city by defining an optimal mix of crops. Its objective is to increase expected income while simultaneously reducing exposure to risk, measured through the minimisation of the Mean Absolute Deviation.

Model elements

The decision framework presented in this research focuses on the selection of crop types and the allocation of land among them from the perspective of a small-scale farmer. The objective is to ensure income stability by simultaneously maximising total income and minimising the risk associated with price volatility and operating costs. Model parameters were defined based on a combination of empirical data, regional agricultural characteristics, and technical guidelines, in order to ensure consistency with the production conditions faced by smallholders.

Specifically, the set of crops considered in the model was identified and prioritised in Phase 2 of the research through an evaluation of locally relevant crops, informed by regional production statistics and expert input. Minimum and maximum land allocation levels for each crop were defined according to regional agronomic recommendations, typical farm sizes, and land-use practices observed among small-scale farmers in the study area. These bounds reflect practical limitations related to labour availability, crop management capacity, and land fragmentation.

It is assumed that farmers generate income from the initial periods of the planning horizon because of ongoing production cycles from previously planted crops, reflecting the continuous nature of agricultural activities. Harvest seasonality is incorporated through crop-specific parameters that activate production in predefined time intervals, based on optimal planting windows and crop cycle durations derived from regional agronomic calendars.

The planning horizon is defined monthly, and historical price data from the preceding 15 months are used to characterise price variability for all crops. Finally, farmer income is projected over a two-year period (24 months), which is consistent with medium-term planning practices in small-scale agriculture and allows the model to capture income dynamics across multiple production cycles.

Basic notation

This section provides a detailed overview of the sets, parameters, and variables that are common to the formulation of all the models presented.

Sets

$\:C:Set\:of\:Crops,\:indexed\:in\:c$

$\:T:Set\:of\:future\:periods\:within\:the\:land\:planning\:horizon,indexed\:in\:t$

$\:P:Set\:of\:historical\:periods\:used\:for\:parameter\:estimation,indexed\:in\:p$

Parameters

The parameters include fundamental ones related to each crop and the available land for planting

$\:{rend}_{ct}=Yield\:in\:tons\:per\:hectare\:of\:crop\:c\:in\:period\:t\:assuming\:that\:the\:crop\:is\:$

$\:sown\:in\:the\:optimal\:cycle\:for\:harvesting.$

$\:{tc}_{ct}=1\:if\:crop\:c\:is\:harvested\:in\:period\:t,0\:otherwise,\:considering\:that\:the\:$

$\:sowing\:is\:done\:in\:the\:optimal\:cycle\:for\:harvesting\:$

$\:{Hmin}_{c}=Minimum\:land\:in\:hectares\:to\:be\:planted\:with\:the\:crop\:c$

$\:{Hmax}_{c}=Maximum\:land\:in\:hectares\:to\:be\:planted\:with\:the\:crop\:c$

$\:HD=Land\:available\:for\:planting$

The parameters associated with the behaviour of crop prices are also considered.

$\:{r1}_{cpt}=Given\:a\:future\:period\:t,\:this\:parameter\:represents\:the\:price\:of\:crop\:c$

$\:in\:the\:same\:period\:p=t\:but\:from\:previous\:years.$

$\:ipp=annual\:producer\:price\:index$

$\:td=monthly\:discount\:rate$

$\:{r2}_{cpt}=Price\:of\:crop\:c\:in\:the\:future\:period\:t\:based\:on\:its\:past\:price$

$\:{r2}_{cpt}={r1}_{ctp}*{(1+ipp)}^{(\left|P\right|-(p-⌈\frac{t}{12}⌉\left)\right)}$

$\:{r}_{ct}=Average\:price\:of\:crop\:c\:for\:future\:period\:t$

$\:{r}_{ct}=\frac{1}{\left|P\right|}*\sum\:_{p\:\in\:P}{r2}_{cpt}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(14\right)$

$\:{a}_{ctp}=Distance\:from\:the\:price\:of\:crop\:c\:updated\:in\:period\:t\:to\:the\:average\:$

$\:price\:of\:crop\:c\:in\:the\:same\:period\:t\:\:$

$\:{a}_{ctp}={r2}_{cpt}-{r}_{ct}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(15\right)$

Variables

Among the variables considered in the model are the key decisions made by small-scale producers

$\:{X}_{c}=Amount\:of\:land\:planted\:with\:the\:crop\:c$

$\:{W}_{c}=1\:if\:crop\:c\:is\:sown,0\:otherwise\:in\:its\:optimal\:planting\:cycle$

Income Maximisation Model (IM)

The first model is formulated as a Mixed-Integer Linear Programming (MILP) problem that seeks to maximise the income generated by the association over the entire planning horizon. This objective reflects the fact that smallholder farmers depend directly on the income derived from agricultural activities both to sustain their livelihoods and to reinvest in subsequent planting and harvesting operations. Given their limited financial capacity, ensuring sufficient income is critical to the continuity of production. The detailed formulation of IM model is presented below.

At this model, two additional variables must be incorporated to properly define the objective function.

$\:{INP}_{t}=Farme{r}^{{\prime\:}}s\:income\:in\:future\:period\:t$

$\:INGTOT=Total\:income\:of\:small-scale\:farmers$

The objective function is linked to the maximisation of the farmer's income over the observation period, with the income being discounted to its present value (see Eq. 16)

$\:Max\:Z1=\:\sum\:_{t\:\in\:T}\sum\:_{c\:\in\:C}{r}_{ct}*{rend}_{c}*{X}_{c}*{tc}_{ct}*\frac{1}{{\left(1+td\right)}^{t}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(16\right)$

To simplify the objective function in Eq. 17 the variable

$\:{INP}_{t}$

, is defined, resulting in the expression presented in Eq. 18

$\:{INP}_{t}=\sum\:_{c\:\in\:C}{r}_{ct}*{rend}_{c}\text{*}{X}_{c}\text{*}{tc}_{ct}\text{*}\frac{1}{{\left(1+td\right)}^{t}}\:\forall\:\:t\in\:T\:\:\:\left(17\right)$

$\:Max\:Z1=INGTOT=\:\sum\:_{t}{INP}_{t}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(18\right)$

Constraints

The model has the following basic constraints.

Equations (19–21) show that, if it is decided to plant a crop, the land allocation must range between the maximum and minimum available for each crop. Eq. 16 reflects that no more than the amount of land available to the small farmer can be planted.

$\:{X}_{c}\:\ge\:{Hmin}_{c}\text{*}{W}_{c\:\:\:\:\:\:\:\:\:\:\:}\forall\:\text{c}\:\in\:C\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(19\right)$

$\:{X}_{c}\le\:{Hmax}_{c}\text{*}{W}_{c\:\:\:\:\:\:\:\:\:\:\:}\forall\:\text{c}\:\in\:C\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(20\right)$

$\:\sum\:_{\text{c}\:\in\:C}{X}_{c}\le\:HD\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(21\right)\:$

Likewise, Equations (22–25) define the constraints related to binary decision variables as well as the non-negativity requirements.

$\:{W}_{c\:}\in\:\left\{\text{0,1}\right\}\:\:\:\:\:\:\:\:\forall\:\text{c}\:\in\:C\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(22\right)\:\:\:\:\:$

$\:{X}_{c\:}\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\forall\:\text{c}\:\in\:C\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(23\right)\:\:\:\:\:$

$\:{INP}_{t}\ge\:0\:\:\:\:\:\:\:\:\:\:\:\forall\:\text{t}\:\in\:T\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(24\right)\:\:\:\:\:$

$\:INGTOT\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(25\right)\:\:\:\:\:$

Risk Minimisation Model (RMM)

The second model is formulated as a Mixed-Integer Linear Programming (MILP) problem that aims to minimise the Mean Absolute Deviation (MAD) of incomes across the planning horizon. The rationale is that the crop mix selected by the smallholder association should generate the lowest possible variability in cash flows, thereby enhancing income stability. This is particularly relevant given the inherent variability in crop prices, which introduces uncertainty and risk into production decisions. By minimising the MAD, the model seeks to reduce the farmers’ exposure to such risk. The detailed formulation of RMM model is presented below.

At this model, two additional variables must be incorporated to properly define the objective function.

$\:{Y}_{p}=Absolute\:deviation\:of\:the\:income\:obtained\:by\:the\:crop\:combination\:in\:each\:period$

$\:p\:with\:respect\:to\:the\:best\:historical\:average\:income.$

$\:MADINGR=Average\:absolute\:deviation\:of\:small-scale\:farmers{\prime\:}\:incomes$

The objective function is related to the minimisation of the mean absolute deviation of the farmer's income with respect to the historical average income per period, adapting the approach presented by [35] as shown in Eq. 12. The objective function obtained is presented in Eq. 26.

$\:Min\:Z2=\:\frac{1}{\left|P\right|}\text{*}\sum\:_{p\:\in\:P}\left|\sum\:_{t\in\:T}\sum\:_{c\:\in\:C}{a}_{cpt}\text{*}{X}_{c}\text{*}{rend}_{c}\text{*}{tc}_{ct}\right|\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(26\right)$

Where

$\:{a}_{cpt}$

is defined in Eq. 27, considering that past price behaviour is carried forward to the present.

$\:{a}_{cpt}={r}_{cpt}\text{*}{\left(1+ipp\right)}^{\left(\left|P\right|-\left(p-\left|\frac{t}{12}\right|\right)\right.}-{r}_{ct}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\forall\:c\:\in\:C,\:\forall\:p\:\in\:P,\:\forall\:t\:\in\:T\:\:\:\left(27\right)$

Constraints

The model considers constraints (19–23). To linearize the objective function expressed in Eq. 26, Equations 28, 29 and 30 are proposed. In addition, two constraints are added to ensure the non-negativity of the newly defined variables (see equations 31 and 32).

$\:{Y}_{p}=\:\left|\sum\:_{t\in\:T}\sum\:_{c\:\in\:C}{a}_{cpt}\text{*}{X}_{c}\text{*}{rend}_{c}\text{*}{tc}_{ct}\right|\:\:\:\forall\:p\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(28\right)$

$\:{Y}_{p}+\:\sum\:_{t\in\:T}\sum\:_{c\:\in\:C}{a}_{ctp}{\text{*}X}_{c}\text{*}{rend}_{c}\text{*}{tc}_{ct}\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\forall\:p\in\:P\:\:\:\:\:\:\:\:\:\:\:\:\:\left(29\right)$

$\:{Y}_{p}-\:\sum\:_{t\in\:T}\sum\:_{c\:\in\:C}{a}_{ctp}{\text{*}X}_{c}\text{*}{rend}_{c}\text{*}{tc}_{ct}\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\forall\:p\in\:P\:\:\:\:\:\:\:\:\:\:\:\:\:\left(30\right)$

$\:{Y}_{p}\ge\:0\:\:\:\:\:\:\:\:\:\:\:\forall\:\text{p}\:\in\:P\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(31\right)\:\:\:\:\:$

$\:MADINGR\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(32\right)\:\:\:\:\:$

Considering the linearisation performed, the objective function originally presented in Eq. (26) is redefined as a linear objective function, expressed in Eq. (33).

$\:Min\:Z2=MADINGR=\:\frac{1}{\left|P\right|}\sum\:_{p}{Y}_{p}\:\:\:\:\:\:\:\:\:\:\:\:\:\left(33\right)$

Minimum Income Maximisation Model (MIM)

The third model is formulated as a Mixed-Integer Linear Programming (MILP) problem that seeks to maximise the minimum income per period. This objective reflects the importance, in the context of smallholder farming, of ensuring liquidity throughout the entire planning horizon rather than concentrating it in only a few periods.

By doing so, the model addresses the critical need for small farmers to maintain a minimum income that guarantees both the continuity of agricultural operations and their household subsistence. The detailed formulation of MIM model is presented below.

At this model, one additional variable must be incorporated to properly define the objective function.

$\:INGMIN=minimum\:income\:for\:small-scale\:farmers$

The third objective function implies the maximisation of the minimum income of the small farmer, as shown in Eq. 34

$\:Max\:Z3=INGMIN\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(34\right)$

Constraints

The model considers constraints (19–23). Eq. (35) is employed to determine the minimum cash flow across the entire planning horizon. Likewise, Eq. (36) incorporates a constraint to ensure the non-negativity of the new decision variable.

$\:{INP}_{t}\ge\:INGMIN\:\:\:\:\:\:\:\:\:\:\:\:\:\:\forall\:t\:\in\:T\:\:\:\:\:\:\:\:\:\left(35\right)$

$\:INGMIN\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(36\right)\:\:\:\:\:$

Compromise Portfolio Model (CPM)

The final model is formulated as a Mixed-Integer Linear Programming (MILP) compromise model that simultaneously seeks to maximise total income, maximise the minimum income per period, and minimise risk by reducing the Mean Absolute Deviation (MAD) of income across periods. This formulation integrates multiple decision metrics that are particularly relevant in the context of smallholder agriculture, where it is not only essential to ensure the highest possible income (both cumulative and per period) but also to reduce risk, understood here as the variability of income arising from fluctuations in market prices.

The detailed formulation of CPM model is presented below.

Regarding the decision variables, the formulation incorporates all variables defined in the three preceding models. At this model, one additional variable must be incorporated to properly define the objective function.

$\:NINGTOT=normalised\:total\:income\:for\:small-scale\:farmers$

$\:NMADINGR=normalised\:average\:absolute\:deviation\:for\:small-scale\:farmers$

$\:NINGMIN=normalised\:minimum\:income\:for\:small-scale\:farmers$

$\:BALNOR=normalised\:balance\:across\:all\:decision\:metrics.$

To formulate the objective function, commitment-based programming is used, where the models are run with each of the three objective functions proposed above, also recording the metrics that have not been optimized. For example, the goal is to maximise the small-scale farmer's total income, while recording the mean absolute deviation and the minimum income of the producer during this process, even though these indicators have not been optimised. Following this step, new parameters are created to store the best and worst results for each of the three metrics considered in the study, as outlined below.

$\:pingtot=worst\:total\:farme{r}^{{\prime\:}}s\:income\:metric\:performance$

$\:mingtot=Best\:total\:farme{r}^{{\prime\:}}s\:income\:metric\:performance$

$\:pmading=Worst\:mean\:absolute\:deviation\:in\:the\:{farmer}^{{\prime\:}}s\:income\:metric\:performance\:$

$\:mmadingr=Best\:mean\:absolute\:deviation\:in\:the\:{farmer}^{{\prime\:}}s\:income\:metric\:performance\:$

$\:pingmin=Worst\:minimun\:income\:metric\:performance\:$

$\:mingmin=Best\:minimun\:income\:metric\:performance\:$

Based on the best and worst results for each metric, normalised metrics are constructed for each objective, as shown in equations 37, 38 and 39.

$\:NINGTOT=\:\frac{mingtot\:-INGTOT}{mingtot\:-pingtot}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(37\right)$

$\:NMADINGR=\:\frac{pmadingr-MADINGR}{pmadingr\:-mmadingr}\:\:\:\:\:\:\:\:\:\left(38\right)$

$\:NINGMIN=\:\frac{mingmin\:-INGMIN}{mingmin\:-pingmin}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(39\right)$

Constraints

The model considers constraints (19–25), (29–32), (35–39). Once the normalized metrics are defined, a new variable called

$\:BALNOR$

is created, which aims to find a balance between the three normalized metrics. To achieve this goal, equations 40, 41 and 42 are proposed.

$\:BALNOR\le\:NINGTOT\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(40\right)$

$\:BALNOR\le\:NMADINGR\:\:\:\:\:\:\:\:\:\:\:\:\left(41\right)$

$\:BALNOR\le\:NINGMIN\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(42\right)$

In addition, non-negativity constraints are imposed on the newly defined variables (see equations 43, 44, 45 and 46).

$\:NINGTOT\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(43\right)\:\:\:\:\:$

$\:NMADINGR\:\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(44\right)\:\:\:\:\:$

$\:NINGMIN\:\:\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(45\right)\:\:\:\:\:$

$\:BALNOR\ge\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(46\right)\:\:\:\:\:$

Finally, the objective function of the model (see Eq. 47) is the maximization of

$\:BALNOR$

, since the closer this variable is to 1, the better the balance between the three objective metrics initially defined.

$\:Max\:Z4=BALNOR\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(47\right)$

4. Results

Case study

This research uses as a case study an association of small farmers in a rural area of the municipality of Guadalajara de Buga, in Valle del Cauca, Colombia. This choice is based on the findings of previous studies, such as that of [8], which identified the urgent need for tools to improve crop planting planning.

The objective is to improve the living conditions of this rural population while contributing to the strengthening of the internal food supply of the department of Valle del Cauca - Colombia. The research aims, therefore, to provide practical and effective solutions that boost both the agricultural productivity of small-scale farmers and their socioeconomic well-being, promoting sustainable development in the region.

Phase 1. Characterization of the farmer under study

This research focuses on the department of Valle del Cauca, Colombia, one of the main food pantries in the country. Previous studies, such as [50] have identified the potential of various crops for the region, highlighting their current production and future projections. Based on these crops, a survey was designed in a rural area of a city in Valle del Cauca, where a sample of 90 producer families was obtained. The survey aimed to validate the relevance of each crop and to understand the farmers' preferences regarding what to plant. For this purpose, a Likert scale from 1 to 10 was used, where 1 represents the highest preference and 10 is the lowest.

After consulting primary sources, a consolidated list of 45 possible crops to be planted in rural areas of the department of Valle del Cauca was obtained, distributed among fruits, vegetables, bulbs, tubers, roots and other crops, as shown in Fig. 2.

Fig. 2

Composition of crops suitable for planting in rural areas of Valle-Colombia.

From Fig. 2, it can be observed that fruit crops predominate in the study area, accounting for 49% of the total cultivated area, followed by vegetables with 27%. This distribution indicates a strong reliance on perennial and short-cycle crops, which may influence both the seasonality of income and the exposure of producers to price and climate-related risks.

Phase 2. Prioritization of crops using a multicriteria tool

Along with identifying the crops suitable for planting in the region under study, the review of government technical reports and the surveys conducted in the first phase of the research facilitated the collection of key information on each crop. In this study, the criteria used for prioritisation were based on those previously adopted in the literature, while also considering the availability of information for each criterion with respect to each crop, as this is essential for the application of the TOPSIS method. After contrasting the criteria identified in the literature with the data available, five criteria were defined for this research: four quantitative ones—annual production in tonnes, cost and yield per hectare, and the price received by the producer—and one qualitative criterion, obtained through the survey, reflecting farmers’ preferences for planting specific crops based on their own management experience.

Considering the 45 crops as alternatives and the 5 established criteria, the TOPSIS method is applied. To achieve this, the data obtained for each criterion for each alternative in the first phase of the study are used. Once the decision matrix is constructed, it is normalized according to Eq. 3. The resulting matrix is multiplied by the weight assigned to each criterion, considering that they all have the same weight (20%).

Subsequently, the weighted normalized matrix is obtained and the positive ideal solution (A+) and the negative ideal solution (A-) are defined according to the objective of each criterion. In this case, the criteria of annual production, yield and price paid to the producer will be maximized, while the criteria of preference and cost per hectare will be minimized. It is important to note that, for the preference criterion, the lower the value, the fewer difficulties the farmer faces when planting the crop. Table 1 presents the ideal negative and positive solutions for each criterion.

Table 1

Positive and negative ideal solutions by criterion
	Annual production (ton)	Crop yield (ton/ha)	Price paid to the producer ($/ton)	Cost per hectare of crop (S/Ha)	Farmer’s Preference
A+	0,120164496	0,088799461	0,095778224	0,011080176	0,002245319
A-	7,34781E-06	0,006108247	0,009737453	0,096279266	0,045942379

Next, the distance of each alternative to the ideal negative and positive solution is calculated, using equations 7 and 8. Subsequently, the relative proximity value Ri is determined using Eq. 9. Finally, the alternatives are ordered from highest to lowest Ri value and the 80th percentile is calculated, selecting those crops whose Ri value is higher than this percentile.

The 80th percentile was chosen as the cut-off point to ensure that only alternatives with a relative proximity value (Ri) in the top performance range were retained. This threshold not only prioritises crops with consistently strong performance across the evaluation criteria while excluding less robust options but also prevents the inclusion of an excessive number of crops, which could complicate farm management and result in cultivation areas becoming too small on average.

Since the selection of crops may be influenced by the weighting of the criteria—initially set at 20% each—it is important to ensure that the decision remains robust, i.e. not overly sensitive to variations in the assigned weights. To this end, a sensitivity analysis of the baseline case is conducted. In this analysis, the weight of the qualitative criterion related to farmers’ preferences is systematically increased and decreased, while the remaining weight is redistributed equally among the four quantitative criteria.

The focus on this criterion is deliberate: unlike strictly quantitative measures such as yield, cost, or price, farmers’ preferences are inherently subjective and may vary substantially across contexts and over time. Assessing the impact of such variability on the prioritisation outcome strengthens the robustness and practical reliability of the proposed decision-making framework.

Table 2 presents the crops selected for this study, namely those that appear most frequently above the 80th percentile across the scenarios simulated with TOPSIS. Notably, six crops—pineapple, orange, avocado, Hass avocado, mandarin, and banana—were consistently included in 100% of the scenarios. Furthermore, seven crops (the six mentioned above plus papaya) align with those prioritised by the government of Valle del Cauca, which further reinforces the robustness of their selection for this research.

Table 2

Crops prioritised using the TOPSIS method
Crop	Ri – Initial case	% of scenarios in which crop is prioritised
Orange	0,646080384	100
Pineapple	0,603123055	100
Hass avocado	0,568565774	100
Banana	0,496423738	100
Avocado	0,389817183	100
Papaya	0,328035244	89
Tangerine	0,313378055	100
Beet	0,270127536	56
Mango	0,269497029	78
Lulo	0,266751726	54

Phase 3. Definition of crop portfolio through the mathematical model

The proposed model was implemented in the AMPL language and applied to the case study. It is important to note that the data used for this model are taken from database references of the National Department of Statistics (DANE), technical reports prepared by the Government of Valle del Cauca and previous studies conducted in fresh food supply chains with institutions such as Universidad de los Andes, Universidad Distrital Francisco José de Caldas and Special Administrative and Planning Region (RAP-E) of the central region comprising the departments of Cundinamarca, Boyacá, Meta, Huila and Tolima in Colombia.

Table 3 presents the results of the metrics considered in the study, both when optimised independently and when optimised jointly (as indicated by the BALNOR variable). It shows that the best results for each metric are achieved when optimised individually. Furthermore, it is observed that when maximising the BALNOR variable, an intermediate outcome is obtained across all three metrics, where the results are neither the best nor the worst compared to the other scenarios.

Table 4 reports the results obtained for each crop under the different optimisation metrics considered. When the objective is to minimise the mean absolute deviation (MAD) of income—used here as a proxy for income risk—the model selects only two crops and allocates them in relatively small land areas. This outcome reflects a clear trade-off between diversification and income volatility: although cultivating a larger number of crops may appear to reduce production risk, in this case several crops exhibit highly volatile prices and costs, which increases overall income variability. Consequently, concentrating production on a limited number of relatively stable crops emerges as a risk-minimising strategy for smallholders whose primary concern is income predictability rather than return maximisation.

In contrast, when the optimisation focuses on maximising total income or ensuring the highest minimum income level, the model favours a more diversified crop portfolio, albeit still excluding certain crops whose expected returns do not compensate for their associated costs or volatility. These scenarios illustrate the classical trade-off faced by small-scale farmers between higher expected income and greater exposure to income fluctuations, highlighting how different risk attitudes lead to distinct crop allocation strategies.

Finally, under the normalised balance metric (BALNOR), which jointly considers income level and income stability, the model allocates land to all ten available crops. This result suggests that diversification becomes beneficial when income variability is explicitly balanced against expected returns, rather than minimised in isolation. From a practical perspective, this strategy allows farmers to smooth income over time by combining crops with different harvest cycles and price dynamics, thereby reducing dependence on a limited number of income sources while avoiding excessive exposure to highly volatile crops.

Importantly, across all four scenarios, the model guarantees a minimum income level for smallholders, reinforcing its suitability as a decision-support tool in vulnerable rural contexts. For policymakers and extension services, these findings underline the importance of promoting crop planning strategies that align with farmers’ risk preferences: while highly risk-averse farmers may benefit from simplified crop portfolios, balanced diversification strategies can enhance economic resilience, support more stable employment, and contribute to local food availability when appropriate risk–return trade-offs are considered.

Table 3

Results of metrics considered in the research
Optimized metrics	INGTOT	MADINGR	INGMIN	BALNOR
INGTOT	$ 36.317.400.000	$ 13.747.200.000	$ 635.580.000	0
MADINGR	$ 4.520.970.000	$ 1.684.380.000	$ 152.214.000	0
INGMIN	$ 29.944.200.000	$ 11.288.600.000	$ 1.039.850.000	0
BALNOR	$ 20.421.100.000	$ 7.715.090.000	$ 596.087.000	0,5

Table 4

Crops planted by the small producers' association according to each optimized metric
Crops	MADINGR	INGTOT	INGMIN	BALNOR
1	0	0	7,54	3
2	0	6	3	3
3	0	12	7,22	4,13
4	0	0	7,51	3,8
5	3	10	10	6,1
6	0	6	0	1
7	0	7	3,8	2,4
8	1,38	6	0	4,4
9	0	0	3	2
10	0	3	4,8	3
Total	4,38	50	46,87	32,83

5. Discussion

Several approaches to the crop planning problem have been addressed, including the use of multicriteria tools [51], simulation models based on system dynamics [52], and optimisation-based approaches [19], [22], [53]. Within mathematical modelling, mixed-integer linear programming (MILP) models [16], [17], [54] and multi-objective formulations [23], [28] have been widely used to address the inherent trade-offs in crop planning decisions. In this research, a commitment-based MILP model is developed to jointly determine crop selection and land allocation while explicitly accounting for total income, minimum income, and income variability for an association of small-scale producers.

Crop planning has been extensively studied as a multi-criteria decision problem [26], with prior research incorporating economic [19], [21], environmental [22], [55] and social dimensions [28]. These contributions underscore the importance of integrating profitability, sustainability, and social impact to support efficient and viable agricultural production systems. However, many of these approaches implicitly assume stable or aggregated economic conditions, limiting their ability to reflect the temporal and behavioural dimensions faced by smallholders.

From an economic perspective, most crop planning models prioritise income maximisation [20] or cost minimisation [56]. Although portfolio-based approaches have introduced risk measures to address uncertainty, they typically rely on static or annualised representations of prices and returns and often assume homogeneous risk preferences. As a result, they provide limited insight into how farmers manage short-term income fluctuations that directly affect household liquidity and survival. The risk associated with crop planning decisions therefore remains insufficiently addressed, particularly in contexts characterised by high price volatility [57].

This study contributes to the literature by extending portfolio-based crop planning approaches in two keyways. First, the proposed model incorporates farmers’ risk preferences explicitly through a commitment-based formulation that balances income maximisation with income stability, rather than treating risk solely as a statistical property of returns. Second, monthly price scenarios are integrated into the optimisation framework, allowing the model to capture seasonality, harvest cycles, and short-term market fluctuations that are typically overlooked in static portfolio models. This temporal resolution enables a more realistic representation of farmers’ cash flows and supports decision-making that aligns with household-level financial commitments.

From a practical perspective, these enhancements translate into actionable benefits for small-scale farmers and policymakers. By identifying crop portfolios that stabilise income across the planning horizon, the model supports farmers in managing liquidity constraints and reducing exposure to adverse price shocks. At the institutional level, the framework provides extension services and local authorities with a transparent tool to evaluate alternative crop planning strategies under different risk preferences and market conditions, thereby facilitating more informed land-use and resource allocation decisions.

Although this study explicitly accounts for price-related risk, further research is required to incorporate additional sources of uncertainty, particularly climate variability. Integrating stochastic representations of rainfall, temperature extremes, and drought conditions would further strengthen the robustness of crop planning decisions and enhance resilience under climate change [55], [58], [59]

6. Theoretical and practical implications

This research contributes to the literature on decision-making in rural communities, particularly those engaged in agricultural activities, by extending portfolio theory to the crop planning problem under income uncertainty. The use of portfolio theory provides a solid foundation for developing models that not only focus on maximising income from multiple crops but also explicitly integrate risk as a key decision-making dimension. This is especially relevant in agricultural contexts characterised by high levels of price volatility and uncertainty.

By adopting a portfolio perspective, this research conceptualises crop diversification primarily as an economic risk-management strategy. Nevertheless, this concept could be further expanded to support adaptive responses to climate change and other external shocks. Integrating diversification with environmental and social dimensions would enhance the understanding of agricultural system resilience, allowing uncertainty to be managed more holistically rather than solely from an economic standpoint.

From a practical perspective, the proposed model can support farmers by enabling them to identify crop portfolios that balance income generation with income stability. One potential application is the development of decision-support software that allows farmers and extension agents to simulate alternative crop plans under different price scenarios, resource constraints, and risk preferences, thereby facilitating more informed and transparent decision-making.

Despite these contributions, several limitations should be acknowledged. First, the model relies on historical price data to generate monthly scenarios, which may not fully capture structural market changes or extreme events that deviate from past patterns. Second, data granularity remains a constraint, as price and cost information is typically aggregated at regional or monthly levels, potentially overlooking local market heterogeneity. Finally, the model assumes rational adoption of recommendations, while behavioural factors—such as risk perception, tradition, or limited trust in analytical tools—may influence farmers’ willingness to implement diversification strategies in practice.

Regarding policy implications, the results suggest that crop diversification policies should be tailored to farmers’ risk profiles rather than promoted uniformly. Public interventions could include subsidy schemes that prioritise crop combinations contributing to income stability, rather than rewarding output volume alone. Additionally, extension services could leverage the proposed model to provide customised advice to smallholders, translating technical optimisation results into practical recommendations aligned with local conditions. Such targeted support mechanisms would enhance the effectiveness of diversification policies and contribute to mitigating income volatility in rural communities.

7. Conclusions and future research

This research proposes a model that integrates portfolio theory into agricultural planning, optimising farming decisions not only for income maximisation but also for risk management. The application of this theory offers a valuable tool for addressing the uncertainty inherent in agricultural activities, enhancing farmers' decision-making in the context of high market price volatility.

The research reaffirms the relevance of interdisciplinary approaches by integrating concepts from financial theory with agricultural economics. The proposed model employs a hybrid methodology that combines the use of a multi-criteria tool with the construction of a mathematical model, which enables a more comprehensive understanding of decision-making in rural communities. This approach not only addresses economic aspects but also incorporates farmers' previous experiences with crops, which contributes to a more comprehensive management of uncertainty.

The model relies heavily on historical and context-specific data, particularly with respect to prices and yields. In rural areas, such data may be limited, outdated, or not fully representative, which could affect the robustness of the results. Although the model accounts for price volatility and seasonal variability, it simplifies the complexity of agricultural systems by not explicitly incorporating external shocks such as pest outbreaks, extreme weather events, or changes in agricultural policy.

Future research could focus on expanding the concept of diversification explored in this study by incorporating deeper social and environmental dimensions. For instance, it could investigate how diversification decisions impact the social cohesion of rural communities, the sustainable management of natural resources, and the mitigation of climate change effects. This broader approach would provide a more comprehensive understanding of the multifaceted benefits of diversification in agricultural systems.

Similarly, in the future, it would be relevant to investigate how portfolio models can be applied not only at the individual level but also at the collective level, promoting the resilience of farmer associations. This would involve analysing how cooperation among farmers to diversify crops and share resources can reduce risks and enhance economic stability at the community level.

Data Availability

Data are available on request.

Acknowledgment

This work was supported by the Ministerio de Ciencia, Tecnología e Innovación (MINCIENCIAS) of Colombia and by the Universidad del Valle, Cali, Colombia.

Disclosure

statement

No potential conflict of interest was reported by the author(s).

Funding

This work was financially supported by Ministerio de Ciencia, Tecnología e Innovación (MINCIENCIAS) of Colombia

Appendix A.

Step 1. Generate a decision matrix based on the performance ratings assigned to each alternative in relation to each attribute, as presented in Eq. 1

$\:D=\:{\left[{X}_{ij}\right]}_{mxn}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(1\right)$

Step 2. Select the importance weight of each attribute (Wj), as provided by the experts, such as (see Eq. 2)

$\:\sum\:_{j=1}^{n}Wj=1\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(2\right)$

Step 3. Normalise the decision matrix to convert the different attribute dimensions into comparable ones, as shown in Eq. 3:

$\:{r}_{ij}=\:\frac{{X}_{ij}}{\sqrt{{\sum\:}_{i=1}^{m}{\left({X}_{ij}\right)}^{2}}}\:,\:i=\text{1,2},\dots\:.m;j=\text{1,2},\dots\:n\:\:\:\:\:\:\:\:\:\:\:\left(3\right)$

Step 4. Construct the weighted normalised decision matrix, as shown in Eq. 4:

$\:V=\:{\left[{V}_{ij}\right]}_{mxn}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left(4\right)$

Where

$\:{V}_{ij}=\:{W}_{ij}\text{*}{r}_{ij},\:i=\text{1,2},\dots\:m;j=\text{1,2},\dots\:n$

Step 5. Determine the positive ideal solution (+) and the negative ideal solution (−), as shown in Equations 5 and 6

$\:{A}^{+}=\left({{V}_{1}}^{+},\:{{V}_{2}}^{+},\dots\:,\:{{V}_{n}}^{+}\right)=\:〈\left(\text{max}{V}_{ij}\left|j\:\in\:\:{\phi\:}_{B}\right.\right),\:\text{m}\text{i}\text{n}\left({V}_{ij}\left|j\:\in\:\:{\phi\:}_{B}\right.\right)〉\:\:\:\:\:\:\:\left(5\right)\:$

$\:{A}^{-}=\left({{V}_{1}}^{-},\:{{V}_{2}}^{-},\dots\:,\:{{V}_{n}}^{-}\right)=\:〈\left(\text{min}{V}_{ij}\left|j\:\in\:\:{\phi\:}_{B}\right.\right),\:\text{m}\text{a}\text{x}\left({V}_{ij}\left|j\:\in\:\:{\phi\:}_{c}\right.\right)〉\:\:\:\:\:\:\:\:\left(6\right)\:$

where

$\:{\phi\:}_{B}$

and

$\:{\phi\:}_{C}\:$

are associated with cost and benefit attribute sets.

Step 6. Calculate the Euclidean distance of each alternative from both the negative and positive ideal solutions, as shown in Equations 7 and 8

$\:{{d}_{i}}^{+}=\:\sqrt{{\sum\:}_{j=1}^{n}{({V}_{ij}-{{V}_{j}}^{+})}^{2}}$

, i = 1, 2…, m (7)

$\:{{d}_{i}}^{-}=\:\sqrt{{\sum\:}_{j=1}^{n}{({V}_{ij}-{{V}_{j}}^{-})}^{2}}$

, i = 1, 2…, m (8)

Step 7. Calculate the proximity coefficient to the ideal solution (Ri) for each alternative, as shown in Eq. (9)

$\:{R}_{i}=\:\frac{{{d}_{i}}^{-}}{{{d}_{i}}^{-}+{{d}_{i}}^{+}},\:0\le\:Ri\le\:1,\:i=\text{1,2},\dots\:m\:\:\:\:\:\:\:\:\:\:\left(9\right)$

Step 8. Determine the ranking preference order of the alternatives in descending order, with alternatives that are closer to the positive ideal solution and farther from the negative ideal solution ranked higher.

Author Contribution

AMPR: Writing – review & editing, Writing – original draft, Methodology, Investigation, Formal analysis, Conceptualization. JPOC: Supervision, Project administration, Methodology, Investigation, Conceptualization. JCOG: Validation, Project administration, Formal analysis, Conceptualization.

References

Orjuela-Castro JA, Orejuela-Cabrera JP, Adarme-Jaimes W Supply Chain Network Design of Perishable Food in SurplusPeriods, Int. J. Food Syst. Dyn., vol. 14, no. 1, pp. 110–127, 2023, [Online]. Available: https://dx.doi.org/10.18461/ijfsd.v14i1.E8

Clavijo-Buritica N, Triana-Sanchez L, Escobar JW (2023) A hybrid modeling approach for resilient agri-supply network design in emerging countries: Colombian coffee supply chain. Socioecon Plann Sci. 10.1016/j.seps.2022.101431

Falkowski J (2015) Resilience of farmer-processor relationships to adverse shocks: the case of dairy sector in Poland. Br Food J 117(10):2465–2483

Accorsi R, Gallo A, Manzini R (2017) A climate driven decision-support model for the distribution of perishable products. J Clean Prod 165:917–929. 10.1016/j.jclepro.2017.07.170

Gholami-Zanjani SM, Klibi W, Jabalameli MS, Pishvaee MS (2021) The design of resilient food supply chain networks prone to epidemic disruptions. Int J Prod Econ 233. 10.1016/j.ijpe.2020.108001

DANE (2019) Boletín Técnico. Encuesta Nacional Agropecuaria (ENA)

DANE (2025)

Peña-Orozco DL, Londoño-Escobar ME, Paredes AM, Rodríguez J, Gonzalez-Feliu, Navarrete Meneses G (2024) Prioritizing Public Policy Implementation for Rural Development in a Developing Country via Multicriteria Classification. Economies 12(1):1–25. 10.3390/economies12010003

Peña-Orozco DL, Desgourdes C, Gonzalez-Feliu J, Velez-Osorio E, Cardona-Muñoz JD (2024) Harvest planning for efficient and integrated food supply chains: a combined project-operations management approach. Supply Chain Forum Int J 1–15. 10.1080/16258312.2024.2433940

10.

Chen T et al (2024) Optimizing path planning for a single tracked combine harvester: A comprehensive approach to harvesting and unloading processes. Comput Electron Agric 224. 10.1016/j.compag.2024.109217

11.

Sun J et al (2024) Fruit flexible collecting trajectory planning based on manual skill imitation for grape harvesting robot. Comput Electron Agric 225. 10.1016/j.compag.2024.109332

12.

Perondi D et al (2018) Crop season planning tool: Adjusting sowing decisions to reduce the risk of extreme weather events, Comput. Electron. Agric., vol. 156, no. November pp. 62–70, 2019. 10.1016/j.compag.2018.11.013

13.

Omotayo AO, Aremu AO (2024) How does the decision to cultivate underutilized crops influence food and nutrition security in rural areas? J Agric Food Res 18. 10.1016/j.jafr.2024.101400

14.

Rǎdulescu M, Rǎdulescu CZ, Zbǎganu G (2014) A portfolio theory approach to crop planning under environmental constraints. Ann Oper Res 219(1):243–264. 10.1007/s10479-011-0902-7

15.

Filippi C, Mansini R, Stevanato E (2017) Mixed integer linear programming models for optimal crop selection. Comput Oper Res 81:26–39. 10.1016/j.cor.2016.12.004

16.

Galán-Martín Á, Pozo C, Guillén-Gosálbez G, Antón A, Vallejo, Jiménez Esteller L (2015) Multi-stage linear programming model for optimizing cropping plan decisions under the new Common Agricultural Policy, Land use policy, vol. 48, no. pp. 515–524, 2015. 10.1016/j.landusepol.2015.06.022

17.

Alemany MME, Esteso A, Ortiz Á, Pino Mdel (2020) Centralized and distributed optimization models for the multi-farmer crop planning problem under uncertainty: Application to a fresh tomato Argentinean supply chain case study, Comput. Ind. Eng., vol. 153, no. December 2021. 10.1016/j.cie.2020.107048

18.

Wu L, Tian J, Liu Y, Jiang Z (2022) Multi-Objective Crop Planting Structure Optimisation Based on Game Theory. Water (Switzerland) 14(13). 10.3390/w14132125

19.

Ramos FL, Batres R, De-la-Cruz-Márquez CG, Anzures ML (2023) Optimization models for nopal crop planning with land usage expansion and government subsidy. Socioecon Plann Sci 89. 10.1016/j.seps.2023.101693

20.

Benini M, Blasi E, Detti P, Fosci L (June, 2023) Solving crop planning and rotation problems in a sustainable agriculture perspective. Comput Oper Res 159. 10.1016/j.cor.2023.106316

21.

Wu H, Li X, Lu H, Tong L, Kang S (2023) Crop acreage planning for economy- resource- efficiency coordination: Grey information entropy based uncertain model. Agric Water Manag 289. 10.1016/j.agwat.2023.108557

22.

Debnath M, Sarma AK, Mahanta C (2024) Optimizing crop planning in the winter fallow season using residual soil nutrients and irrigation water allocation in India. Heliyon 10(7). 10.1016/j.heliyon.2024.e28404

23.

Pontes RDSG, Brandão DN, Usberti FL, De Assis LS (2024) Multi-objective models for crop rotation planning problems. Agric Syst 219. 10.1016/j.agsy.2024.104050

24.

Benini M, Detti P, Nerozzi L (2025) Optimization models and algorithms for sustainable crop planning and rotation: An arc flow formulation and a column generation approach. Omega (United Kingdom) 135:103320. 10.1016/j.omega.2025.103320

25.

Hou Y, Liu Y, Xu X, Fan Y, Ma S, Wang S (2023) Grid-scale crop dynamic layout optimization model considering stakeholders’ cropping preferences and practice behaviours. Ecol Indic 155. 10.1016/j.ecolind.2023.110963

26.

Randall M, Schiller K, Lewis A, Montgomery J, Alam MS (2024) A Systematic Review of Crop Planning Optimisation Under Climate Change. Water Resour Manag 38(6):1867–1881. 10.1007/s11269-024-03758-3

27.

Gong Y et al (2025) A mixed integer linear programming framework for mitigating enteric methane emissions on dairy farms through optimized crop and diet planning. J Clean Prod 511:145636. 10.1016/j.jclepro.2025.145636

28.

Esteso A, Alemany MME, Ortiz Á, Iannacone R (2022) Crop planting and harvesting planning: Conceptual framework and sustainable multi-objective optimization for plants with variable molecule concentrations and minimum time between harvests. Appl Math Model 112:136–155. 10.1016/j.apm.2022.07.023

29.

Srivastava P, Singh RM (2017) Agricultural Land Allocation for Crop Planning in a Canal Command Area Using Fuzzy Multiobjective Goal Programming. J Irrig Drain Eng 143(6):1–9. 10.1061/(asce)ir.1943-4774.0001175

30.

Adeyemo J, Otieno F (2010) Differential evolution algorithm for solving multi-objective crop planning model. Agric Water Manag 97(6):848–856. 10.1016/j.agwat.2010.01.013

31.

Lin CC, Deng DJ, Kang JR, Liu WY (2021) A Dynamical Simplified Swarm Optimization Algorithm for the Multiobjective Annual Crop Planning Problem Conserving Groundwater for Sustainability. IEEE Trans Ind Inf 17(6):4401–4410. 10.1109/TII.2020.3029258

32.

Li C et al (2025) Scientific planning of dynamic crops in complex agricultural landscapes based on adaptive optimization hybrid SA-GA method. Sci Rep 15(1):1–22. 10.1038/s41598-025-14188-5

33.

Karthick T, Devi TK, Ramprasath M (2020) Webology 17(2):677–693. 10.14704/WEB/V17I2/WEB17060. Effective Use of Big Data Planning and Analytics in Crop Planting

34.

Gawade SD, Bhansali A, Chopade S, Kulkarni U (2026) Optimizing crop yield prediction with R2U-Net-AgriFocus: A deep learning architecture with leveraging satellite imagery and agro-environmental data, Expert Syst. Appl., vol. 296, no. PB, p. 128942. 10.1016/j.eswa.2025.128942

35.

Konno H, Yamazaki H (1991) Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market. Manage Sci 37(5):519–531. https://doi.org/10.1287/mnsc.37.5.519

36.

Chari A et al (2023) Analysing the antecedents to digital platform implementation for resilient and sustainable manufacturing supply chains - An IDEF0 modelling approach. J Clean Prod 429. 10.1016/j.jclepro.2023.139598

37.

Hajiaghaei-Keshteli M, Cenk Z, Erdebilli B, Selim Y, Özdemir, Gholian-Jouybari F (2023) Pythagorean Fuzzy Topsis Method for Green Supplier Selection in the Food Industry. Expert Syst Appl 224. 10.1016/j.eswa.2023.120036

38.

Zhang T, Feng Q (2024) Multidimensional Risk Assessment of China’s Grain Supply Chain with Entropy Weight TOPSIS Method. Procedia Comput Sci 242:853–858. 10.1016/j.procs.2024.08.213

39.

Najafi M, Zolfagharinia H (2024) A Multi-objective integrated approach to address sustainability in a meat supply chain. Omega 124. 10.1016/j.omega.2023.103011

40.

Cruz NN, Sarmento FJ, Tahimi A, Duarte AD (2025) Implementation of the Balanced Decision-Making Method for Prioritizing Crop Planting in Small-Scale Rural Properties. Pesqui Operacional 45. 10.1590/0101-7438.2025.045.00288556

41.

Ayvaz-Çavdaroǧlu N (2021) A multiple criteria decision analysis for agricultural planning of new crop alternatives in Turkey. J Intell Fuzzy Syst 40(6):10737–10749. 10.3233/JIFS-201701

42.

Chud-Pantoja VL, Manyoma-Velasquez PC, Soto-Paz J (2025) Multicriteria Methodology for Selecting Agrifood Supply Chains With the Potential for Circular Economy Implementation, J. Eng. (United Kingdom), vol. no. 1, 2025. 10.1155/je/5822783

43.

Yoon KP, Hwang CL (1995) Multiple attribute decision making: an introduction

44.

Markowitz HM, Selection P (1952) J Finance 7(1):77–91. 10.1111/j.1540-6261.1952.tb01525.x

45.

Ledoit O, Wolf M (2025) Markowitz Portfolios under Transaction Costs. Q Rev Econ Financ 100. 10.1016/j.qref.2025.101962

46.

García-Cáceres RG, Páez-Rivera FI, Aldana-Gómez B, Acosta-Gempeler E, Escobar-Velásquez JW (2024) An approach to the integral optimization of investment portfolios. J Open Innov Technol Mark Complex 10(1). 10.1016/j.joitmc.2024.100235

47.

Mansini R, Ogryczak W, Speranza MG (2003) LP solvable models for portfolio optimization: A classification and computational comparison. IMA J Manag Math 14(3):187–220. 10.1093/imaman/14.3.187

48.

Simaan Y (1997) Estimation risk in portfolio selection: The mean variance model versus the mean absolute deviation model. Manage Sci 43(10):1437–1446. 10.1287/mnsc.43.10.1437

49.

Li QQ, Li YP, Huang GH, Wang CX (2018) Risk aversion based interval stochastic programming approach for agricultural water management under uncertainty. Stoch Environ Res Risk Assess 32(3):715–732. 10.1007/s00477-017-1490-0

50.

Ministerio de agricultura y desarrollo rural (2020) Priorización de alternativas productivas y diagnóstico del mercado agropecuaria para el departamento del Valle del Cauca. pp. 1–215

51.

Seyedmohammadi J, Sarmadian F, Jafarzadeh AA, Ghorbani MA, Shahbazi F (2018) Application of SAW, TOPSIS and fuzzy TOPSIS models in cultivation priority planning for maize, rapeseed and soybean crops. Geoderma 310:178–190. 10.1016/j.geoderma.2017.09.012

52.

Taheri N, Pishvaee MS, Jahani H (2025) A robust multi-objective optimization model for grid-scale design of sustainable cropping patterns: A case study. Comput Ind Eng 200. 10.1016/j.cie.2024.110772

53.

Caicedo Solano NE, García Llinás GA, Montoya-Torres JR, Ramirez LE, Polo (2020) A planning model of crop maintenance operations inspired in lean manufacturing. Comput Electron Agric 179(May). 10.1016/j.compag.2020.105852

54.

Fikry I, Gheith M, Eltawil A (2021) An integrated production-logistics-crop rotation planning model for sugar beet supply chains. Comput Ind Eng 157. 10.1016/j.cie.2021.107300

55.

da Silva ND, de Oliveira AS (2024) Impact of climatic trends on the water demand and irrigation planning for annual crops in the São Francisco River basin, Brazil. Phys Chem Earth 136. 10.1016/j.pce.2024.103779

56.

Rajakal JP, Tan RR, Andiappan V, Wan YK, Pang MM (2021) Does age matter? A strategic planning model to optimise perennial crops based on cost and discounted carbon value. J Clean Prod 318. 10.1016/j.jclepro.2021.128526

57.

Kabir MJ, Gaydon DS, Cramb R (2025) Evaluation of crop and pond-deepening adaptations to climate change in saline coastal Bangladesh: Benefit-cost and risk analysis. Agric Water Manag 308. 10.1016/j.agwat.2024.109274

58.

Fereidoon M, Koch M (2018) SWAT-MODSIM-PSO optimization of multi-crop planning in the Karkheh River Basin, Iran, under the impacts of climate change. Sci Total Environ 630:502–516. 10.1016/j.scitotenv.2018.02.234

59.

Yoder L, Wardropper C, Irvine R, Harden S (2025) Cover crops as climate insurance: Exploring the role of crop insurance discounts to promote climate adaptation and mitigate risk. J Environ Manage 373. 10.1016/j.jenvman.2024.123506

Yes